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G-index, topological dynamics and the marker property

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Abstract

Given an action of a finite group G on a topological space, we can define its index. The G-index roughly measures a size of the given G-space. We explore connections between the G-index theory and topological dynamics. For a fixed-point free dynamical system, we study the ℤp-index of the set of p-periodic points. We find that its growth is at most linear in p. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.

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References

  1. M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Mathematical Physics, Analysis and Geometry 2 (1999), 323–415.

    Article  MathSciNet  MATH  Google Scholar 

  2. Y. Gutman, Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society 111 (2015), 831–850.

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Gutman, Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory and Dynamical System 37 (2017), 512–538.

    Article  MathSciNet  MATH  Google Scholar 

  4. Y. Gutman, E. Lindenstrauss and M. Tsukamoto, Mean dimension of ℤk-actions, Geometric and Functional Analysis 26 (2016), 778–817.

    Article  MathSciNet  MATH  Google Scholar 

  5. Y. Gutman, Y. Qiao and M. Tsukamoto, Application of signal analysis to the embedding problem of ℤk-actions, Geometric and Functional Analysis 29 (2019), 1440–1502.

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Gutman and M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, Inventiones Mathematicae 221 (2020), 113–166.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Institut des Hautes Études Scientifiques. Publications Mathématiques 89 (1999), 227–262.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Lindenstrauss and M. Tsukamoto, Double variational principle for mean dimension, Geometric and Functional Analysis 29 (2019), 1048–1109.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel Journal of Mathematics 115 (2000), 1–24.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Matoušek, Using the Borsuk—Ulam Theorem, Universitext, Springer, Berlin, 2003.

    MATH  Google Scholar 

  11. R. Shi, Finite mean dimension and marker property, https://arxiv.org/abs/2102.12197.

  12. R. Shi and M. Tsukamoto, Divergent coindex sequence for dynamical systems, Journal of Topology and Analysis, to appear, https://arxiv.org/abs/2103.11654.

  13. M. Tsukamoto, Double variational principle for mean dimension with potential, Advances in Mathematics 361 (2020), Article no. 106935.

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Authors and Affiliations

  1. Department of Mathematics, Kyushu University, Moto-oka 744, Nishi-ku, Fukuoka, 819-0395, Japan

    Masaki Tsukamoto & Mitsunobu Tsutaya

  2. Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan

    Masahiko Yoshinaga

Authors
  1. Masaki Tsukamoto
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  2. Mitsunobu Tsutaya
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  3. Masahiko Yoshinaga
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Corresponding author

Correspondence to Masaki Tsukamoto.

Additional information

Dedicated to Professor Benjamin Weiss on the occasion of his 80th birthday

M. Tsukamoto was supported by JSPS KAKENHI JP18K03275.

M. Tsutaya was supported by JSPS KAKENHI JP19K14535.

M. Yoshinaga was supported by JSPS KAKENHI JP18H01115.

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Tsukamoto, M., Tsutaya, M. & Yoshinaga, M. G-index, topological dynamics and the marker property. Isr. J. Math. 251, 737–764 (2022). https://doi.org/10.1007/s11856-022-2433-0

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  • DOI: https://doi.org/10.1007/s11856-022-2433-0

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